Femtosecond optical parametric generation ofnoncollinear phase matching for a biaxial crystal

Wei Quan Zhang

In an optical parametric generation of a femtosecond pulse for a biaxial crystal, the interaction of threewaves can be used as a model of noncollinear phase matching in which the group velocities of the interactingpulses are suitably linked to each other. For satisfaction of group-velocity matching, the tunable para-metric generation of femtosecond pulses must use noncollinear phase matching. We consider three con-ditions of group-velocity matching for femtosecond pulses. Signal and idler pulses can be obtained whenthe coupled-wave equations, including the group-velocity mismatch and group-velocity dispersion effects,are solved. A Fourier method is an effective method for solving the equations, and from the solution of theequations the relation between duration of output pulses and wavelengths can be obtained. In a compar-ison of collinear and noncollinear matches, when the latter is group-velocity matched, the duration of itsoutpulses are smaller, and the outpulses can be continually tuned from the visible to the mid-infrared.© 2003 Optical Society of America

OCIS codes: 170.5280, 310.2790, 310.3840.

1. Introduction

Femtosecond pulses tunable from the visible to themid-infrared are useful for many applications, suchas �nonlinear� pulse propagation in optical fibers,spectrum technology of higher resolving power, car-rier dynamic studies in semiconductors, the observa-tion of transient chemical species, and laserchemistry. Sources based on optical parametricconversion–amplification have thus received greatattention. Femtosecond optical parametric genera-tion can obtain a tunable femtosecond coherentsource. Danielius et al.1 studied femtosecond opticalparametric generation for a uniaxial crystal beta-barium borate. We will discuss femtosecond opticalparametric generation for a biaxial crystal.

To obtain chirped-pulse amplification techniques forefficient energy exchange, it is necessary that thephase matching �PM� and the group-velocity �GV�matching are satisfied simultaneously. At collinearpropagation, for certain pump wavelengths, only a sig-nal wavelength can satisfy the two matching condi-tions simultaneously. Hence this generation ofcollinear PM cannot be tuned. A decrease in signal

W. Q. Zhang �mike@hosonic.com.cn� is with the Department ofPhysics, Zhejiang Institute of Science and Technology, Hanghou,Zhjiang 310033, China.

Received 19 March 2003; revised manuscript received 11 June2003.

0003-6935�03�275596-06$15.00�0© 2003 Optical Society of America

5596 APPLIED OPTICS � Vol. 42, No. 27 � 20 September 2003

and idler GVs, in the direction of the pump, so as tosynchronize all pulses along the pump wave vector KP,can be obtained by adoption of noncollinear PMgeometries.2–5 It can obtain a tunable source of ul-trashort pulses. Andreoni and Bondani6 studied GVcontrol in the mixing of three noncollinear phase-matched waves for a uniaxial crystal.

KTP is a widely used material for optical paramet-ric generation. Zhang7,8 studied the optical para-metric generation and the femtosecond second- andthird-harmonic light generation in KTP at collinearpropagation. In this paper, first, we calculate thenoncollinear PM and the GV-matching conditions onan arbitrary wave-vector direction for a biaxial crys-tal. The calculations show that PM and GV-matching conditions, suitable for broadbandinteraction, can be simultaneously verified in a num-ber of ways. The corresponding angles for the inter-action waves are calculated. Next, we discuss howthe coupled-wave equations including the GV mis-match and the lowest- and second-order group-velocity dispersions �GVDs� are solved numerically.Last, we get the relations between the pulse durationand wavelengths for a tunable femtosecond pulse.

2. Phase-Matching and Group-Velocity-MatchingConditions

In the principal-axis coordinate, the direction cosinesof the wave vector of the pump light are

k � sin � cos �,

x

kz � cos �,

ky � sin � sin �, (1)

where � is the angle between the wave vector and themain axis z and � is the angle between the projectionof k in the x–y plane and the x axis. The basicequations of a three-wave parametric interaction oc-curring in the geometry depicted in Fig. 1 are theenergy and momentum conservation relations:

1��p � 1��s � 1��i, (2)

Kp � Ks � Ki, (3)

where Kj � 2�nj��j � j � p, s, i�. From Fig. 1 weobtain

Kp � Ks cos �s � Ki cos �i,

Ks sin �s � Ki sin �i, (4)

where Ki � �Ks2 Kp

2 2KpKs cos �s�1�2. We take as

positive the counterclockwise �s angles, with respectto Kp, and clockwise �i angles.

The refractive indices n at the propagation direc-tion are7

n � 21�2�e � A � �b2 � 2bB � A2�2�1�2, (5)

where A � kz2c kx

2a, B � kz2c kx

2a, a � 1�nx2 1�ny

2,b � 1�nx

2 1�nz2, c � 1�ny

2 1�nz2, e � 1�nx

2 1�nz2,

and the or sign corresponds to fast �F� or slow �S�lights, respectively. Assume that all pump, signal,and idler wave vectors lie in same meridian plane �asshown in Fig. 1; the � angle is the same�. The di-rection cosines of s and i wave vectors, respectively,are

kxs � sin�� � �s�cos �, kzs � cos�� � �s�,

kxi � sin�� � �i�cos �, kzi � cos�� � �i�. (6)

For KTP the Sellmeier equations are of the form9

nj � Aj � Bj��1 � Cj�2� � Dj�

2 � j � x, y, z�,(7)

where � is the vacuum wavelength in micrometersand A, B, C, and D are Sellmeier coefficients.

The PM is satisfied in two ways. First, the signaland idler beams are both slow lights �FSS�. Second,

the signal beam is a slow light, and the idler beam isa fast light �FSF� or vice versa �FFS�. The former iscalled type I matching, and the latter is called type IImatching.

When an ultrashort pulse propagates through acrystal, the wave vector K� � can be written as8

K� � � n� 0� 0�c � � ��1� � . . . ,

where ��1� � 1�vg and vg is the GV:

GVj � vg � c��n � �dn�d���0� j � p, s, i�. (8)

We must add the condition of GV matching to theabove system. We considered three conditions of GVmatching for this study: �i� that the component ofGVs parallel to Kp be equal to GVp, i.e.,

cos �s��n � ��dn�d����s � 1��n � ��dn�d����p; (9)

�ii� that the mismatch of GVs,par with respect to GVpbe opposite that of GVi,par, i.e.,

cos �s��n � ��dn�d����s � cos �i��n � �dn�d����i

� 2��n � ��dn�d����p; (10)

and �iii� that the component of GVi parallel to Kp beequal to GVp, i.e.,

cos �i��n � ��dn�d����i � 1��n � ��dn�d����p.(11)

In Eqs. �8�–�11�, dn�d� is8

dn�d� � n3�4�de�d� � dA�d� � H�G�, (12)

where

H � b�db�d� � dB�d�� � Bdb�d� � AdA�d�,

G � �b2 � 2bB � A2�1�2.

For pump and signal lights,

dA�d� � kz2dc�d� � kx

2da�d�, (13)

where da�d� � 2�dnx�d���nx3 2�dny�d���ny

3. Forthe idler light,

dA�d� � kzi2 dc�d� � kxi

2 da�d� � 2kzicdkzi�d�

� 2kxiadkxi�d�, (14)

where from Eqs. �4� and �6� dkzi�d� � sin�� �i�d�i�d�,

dkxi�d� � cos�� � �i�cos �d�i�d�, (15)

d�i�d� � tg�i�1�KsdKs�d� � �KsdKs�d�

� KpdKp�d� � �KsdKp�d�

� KpdKs�d��cos �s���Ks2 � Kp

2

� 2Ks Kp cos �s��. (16)

dB�d�, db�d�, dc�d�, and de�d� have similar formu-las.

Fig. 1. Wave vectors of the three interacting noncollinear phase-matched waves.

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Relations � � ���s� and �s � �s��s� can be foundthat, in principle, provide the values of the pumpingangle � and the incident angle �s at which both PMand GV-matching conditions are fulfilled over thetuning �transparency� range of the crystal.

For the selection of a certain � angle, the set of Eqs.�2�–�7� and Eq. �1� involves, for a given pump wave-length �p, three unknown quantities, namely, �, �s,and �s. From these equations, we get a PM equa-tion. The set of Eqs. �12�–�15� and Eq. �9� �for con-dition �i��, Eq. �10� �for condition �ii��, or Eq. �11� �forcondition �iii�� also involves the three unknown quan-tities. From these equations, we get a GV-matchingequation. We solve numerically the two equationsin term of �s for the selected �p values and find � and�s angles.

3. Phase-Matching Angle

The calculation shows that for type I PM of the KTPcrystal the PM condition and the GV-matching con-dition cannot be satisfied simultaneously.

There are the two kinds of type II PM. Figure 2corresponds to FFS PM �� � 28.6°�. Figure 2�a�shows the dependence of the angle �s values, satis-fying Eq. �9� �condition �i�, solid curves� or Eq. �10��condition �ii�, dashed curves�, on the signal wave-length �s for �p � 395, 527, and 600 nm. Figure2�b� shows the corresponding pump angles �.

Figure 3 corresponds to FSF PM �� � 28.6°�.Figure 3�a� shows the dependence of the angle �svalues, satisfying Eq. �11� �condition �iii�, solidcurves� or Eq. �10� �condition �ii�, dashed curves� onthe signal wavelength �s. Figure 3�b� shows thecorresponding pump angles �. From these figures,we see that the tunable femtosecond pulse can becontinually obtained from the visible to the mid-infrared. At �s � 0.7–1.1 �m, the PM type is FFSPM. At �s � 1.1–2.5 �m, the PM type is FSF PMfor �p � 527 nm. But the tunable curves have agap7 at � � 1.002–1.1007 �m for collinear PM.

4. Coupled Equations Including the Group-VelocityMismatch and the Lowest- and Second-OrderGroup-Velocity Dispersions

The coupled-wave equations of optical parametricgeneration including the GV mismatch and thelowest- and second-order GVDs are given by8,10

dAp�dz � i�2�p�2�d2Ap�d�2 � 1�6�p

�3�d3Ap�d�3

� i8�deff p As� Ai�*exp�i��0 z��c�np, (17)

dAi�dz � ��i�1��cos �idAi�d� � i�2�i

�2��cos �id2Ai�d�2

� 1�6�i�3��cos �id

3Ai�d�3

� i8�deff iAs� Ap�*exp�i��0 z��c�ni, (18)

dAs�dz � ��s�1��cos �sdAs�d�

� i�2�s�2��cos �sd

2As�d�2

� 1�6�s�3��cos �sd

3As�d�3

� i8�deff s Ap Ai exp�i��0 z��c�ns, (19)

where ��i�1� � �i

�1� cos �i �p�1� and ��s

�1� � �s�1� cos

�s �p�1� are the GV mismatches between the idler

and the pump lights and between the signal and the

Fig. 2. �a� Ks-to-Kp angle �s and �b� angle � between Kp and thecrystal’s z axis, as a function of the signal wavelength �s at differ-ent pump wavelengths �p for FFS PM and either GVs,par � GVp

�condition �i�, solid curves� or �GVs,par GVi,par��2 � GVp �condi-tion �ii�, dashed curves�.

Fig. 3. �a� Ks-to-Kp angle �s and �b� angle � between Kp and thecrystal’s z axis, as a function of the signal wavelength �s at differ-ent pump wavelengths �p for FSF PM and either GVi,par � GVp

�condition �iii�, solid curves� or condition �ii� �dotted curves�.

5598 APPLIED OPTICS � Vol. 42, No. 27 � 20 September 2003

pump lights, respectively. ��2� and ��3� are thelowest- and second-order GVD coefficients of theselights, respectively. Their calculations are given inRef. �8�. The PM condition is ��0 � 0. Ap, Ai, andAs are the envelopes of the electric fields of thesewaves. deff is the effective nonlinear coefficient. As-sume that incident pump pulses and signal pulses are

Ap��, 0� � Ap�0���exp���T0� � exp���T0��,

As��, 0� � As�0����exp � ��T0� � exp���T0��. (20)

To use the convolution theory of Fourier transform,we multiply Eqs. �18� and �19� by exp� 2�� ���2�.Then we integrate them over � and carry out a Fou-rier transform. We get

dAi� , z��dz � ���i�1� � i�2�i

�2� 2

� 1�6�i�3� 3��cos �iAi� , z�

� i16�1�2deff i exp�1�4� As� , z�

� F�Ap��, 0��exp� 2�2��c�ni, (21)

dAs� , z��dz � ���s�1� � i�2�s

�2� 2

� 1�6�s�3� 3��cos �s As� , z�

� i16�1�2deff s exp�1�4� Ai� , z�

� F�Ap��, 0��exp� 2�2��c�ns, (22)

where

F�Ap��, 0�� � Ap�0���exp� x� � exp�x��

� x � �T0�2�. (23)

Let

ai � ���i�1�2���T0� � i�i

�2�2x���T0�2

� 1�3�i�3��2x�2���T0�

3��cos �i,

as � ���s�1�2���T0� � i�s

�2�2x���T0�2

� 1�3�s�3��2x�2���T0�

3��cos �s,

bi � 16�1�2deff i exp�1�4�

� F�Ap��, 0��exp� 2�2��c�ni,

bs � 16�1�2deff s exp�1�4�

� F�Ap��, 0��exp� 2�2��c�ns.

The above equations can be written as

dAi� , z��dz � ai� � xAi� , z� � ibi� � As� , z�,

dAs� , z��dz � as� � xAs� , z� � ibs� � Ai� , z�.(24)

Because As� , 0� � As�0���exp�x� exp�x�� andAi� , 0� � 0, solving these equations simultaneously,we can get the Fourier component of the output signalpulse:

As� , L� � 1��2B� As�0���exp� x� � exp�x��

� ��B � �x � as x�exp�BL� � �B � �x

� as x�exp�BL��exp��xL�, (25)

where � � �ai as��2, B � ��ai as�2x2 4bibs�

1�2,and L is the thickness of the crystal. Similarly, wecan get the Fourier component of the output idlerpulse:

Ai� , L� � ibiAs�0���exp� x�

� exp�x����2B�exp��xL�

� �exp�BL� � exp�BL��. (26)

The shapes of the idler and signal pulses can beobtained from the reverse Fourier transform:

Ai��, L� � F1�Ai� , L��, (27)

As��, L� � F1�As� , L��. (28)

At noncollinear matching, for the GV-matching con-dition �i�, ��s � 0, and for the GV-matching condition�iii�, ��i � 0. We calculate the shapes of the idlerpulse for the collinear and the noncollinear matches,respectively.

Figure 4 shows the shapes of the idler pulse atdifferent wavelengths for collinear matching. Fig-ure 5 shows the shapes of the idler pulse at differentwavelengths for noncollinear matching. Figures5�a� and 5�b� correspond to GV-matching condition �i��FFS� and condition �iii� �FSF�, respectively. Figure6 shows the shapes of the idler pulse for noncollinearmatching at GV-matching condition �ii� �FSF�. The

Fig. 4. Idler pulse shapes generated by a 30-fs Nd:YLF pumppulse with crystal thickness L � 0.8 mm. �a� The range �i � 0.7–1�m �corresponding to FSF matching�. P0 is the peak value of thepulse for which �i � 0.7 �m. �b� The range �i � 1.2–1.8 �m�corresponding to FFS matching�. P0 is the peak value of thepulse for which �i � 1.2 �m, at � � 0; for collinear matching, �p �527 nm.

20 September 2003 � Vol. 42, No. 27 � APPLIED OPTICS 5599

thickness of the crystal is 0.8 mm. The peak of elec-tric field intensity of the pump pulse is 1.94 � 106

V�m, and T0 � 30 fs with a wavelength center at 527nm. Figure 7 shows the duration of idler pulses ver-

sus wavelengths for collinear and noncollinearmatches.

From these figures we find that when noncollinearPM and GV-matching conditions �i� or �iii� are satis-fied simultaneously the duration of the idler pulse issmaller than that at collinear matching and that atnoncollinear matching and GV-matching condition�ii�. And the outpulse can be continually tuned.Not only is a shorter pulse duration required for timeresolve, but it is also required to provide sufficientlyhigh peak power.

For noncollinear PM, it is possible to discriminatebetween these matching conditions by rotation of thecrystal so as to obtain the correct pump angle � andthe corresponding �s angle.

5. Conclusion

The GVs of pulses undergoing three-wave parametricinteractions can be calculated in noncollinear PMconditions for a biaxial crystal. We found that thecollinear PM is not a good operation for femtosecondtunable parametric generation because GV matchingand PM cannot be satisfied for any �s simultaneously.For noncollinear PM, the operating conditions are �i�GVp � GVs,par, �ii� GVp � �GVs,par GVi,par��2, or �iii�GVp � GVi,par. For type I PM of the KTP crystal,PM and GV-matching conditions also cannot be sat-isfied simultaneously. For type II at FFS PM, GV-matching conditions �i� or �ii� can be satisfied. In theformer case, GVi,par is left uncontrolled; in the latter,neither GVi,par nor GVs,par is controlled, although theidler and the signal are obviously oppositely mis-matched with respect to the pump. For FSF PM,GV-matching conditions �ii� or �iii� can be satisfied.In the latter, GVs,par is left uncontrolled. By rota-tion of the crystal, so as to obtain the correct pumpangle � and the corresponding �s angle, the tuning ofthe femtosecond parametric generation can beachieved.

In this paper we also developed a method for solv-ing the coupled-wave equations of femtosecond opti-

Fig. 5. Idler pulse shapes generated by a 30-fs Nd:YLF pumppulse with L � 0.8 mm. �a� The range �i � 1.05–1.5 �m �corre-sponding to FFS matching and GV-matching condition �i��. P0 isthe peak value of the pulse for which �i � 1.39 �m. �b� Range �i �0.845–1.05 �m �corresponding to FSF matching and GV-matchingcondition �iii��. P0 is the peak value of the pulse for which �i �0.845 �m; for noncollinear matching, �p � 526 nm at � � 28.6°.

Fig. 6. Idler pulse shapes at �p � 395 nm and GV-matchingcondition �ii�. The �i is 965 nm for FSF matching of � � 28.6°.

Fig. 7. Durations of the idler wave pulses versus wavelengths �i

that include the lowest- and second-order GVDs at noncollinearmatching and GV matching �solid curve� and that include both GVmismatch and the lowest- and second-order GVDs at collinearmatch �dashed curve� for a 30-fs pump pulse.

5600 APPLIED OPTICS � Vol. 42, No. 27 � 20 September 2003

cal parametric generation of noncollinear PM for abiaxial crystal. In the equations the GV mismatchand the lowest- and second-order GVDs must be re-garded. By using the Fourier method and doing anumerical solution, we can solve the equations.Given the pump pulse and the thickness of the crys-tal, the signal and idler pulses can be obtained by thesolution of the equations. We calculate two kinds ofparametric generation of collinear and noncollinear.As GV matching is satisfied at noncollinear type IIPM, the duration of the idler pulse is smaller. It isa optimum operation for tunable femtosecond para-metric generation.

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